Random walks in ( + ) 2 with non-zero drift absorbed at the axes
Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 3, pp. 341-387.

Spatially homogeneous random walks in ( + ) 2 with non-zero jump probabilities at distance at most 1, with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes.

Dans cet article, nous étudions les marches aléatoires du quart de plan ayant des sauts à distance au plus un, avec un drift non nul à l'intérieur et absorbées au bord. Nous obtenons de façon explicite les séries génératrices des probabilités d'absorption au bord, puis leur asymptotique lorsque le site d'absorption tend vers l'infini. Nous calculons également l'asymptotique des fonctions de Green le long de toutes les trajectoires, en particulier selon celles tangentes aux axes.

DOI: 10.24033/bsmf.2611
Classification: 60G50, 60G40, 30E20, 30F10
Keywords: random walk, Green functions, absorption probabilities, singularities of complex functions, holomorphic continuation, steepest descent method
Mot clés : marche aléatoire, fonctions de Green, probabilités d'absorption, singularités de fonctions complexes, prolongement analytique, méthode de la plus grande descente
@article{BSMF_2011__139_3_341_0,
     author = {Kurkova, Irina and Raschel, Kilian},
     title = {Random walks in $(\mathbb {Z}_{+})^{2}$ with non-zero drift absorbed at the axes},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {341--387},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {139},
     number = {3},
     year = {2011},
     doi = {10.24033/bsmf.2611},
     mrnumber = {2869310},
     zbl = {1243.60042},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/bsmf.2611/}
}
TY  - JOUR
AU  - Kurkova, Irina
AU  - Raschel, Kilian
TI  - Random walks in $(\mathbb {Z}_{+})^{2}$ with non-zero drift absorbed at the axes
JO  - Bulletin de la Société Mathématique de France
PY  - 2011
SP  - 341
EP  - 387
VL  - 139
IS  - 3
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/bsmf.2611/
DO  - 10.24033/bsmf.2611
LA  - en
ID  - BSMF_2011__139_3_341_0
ER  - 
%0 Journal Article
%A Kurkova, Irina
%A Raschel, Kilian
%T Random walks in $(\mathbb {Z}_{+})^{2}$ with non-zero drift absorbed at the axes
%J Bulletin de la Société Mathématique de France
%D 2011
%P 341-387
%V 139
%N 3
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/bsmf.2611/
%R 10.24033/bsmf.2611
%G en
%F BSMF_2011__139_3_341_0
Kurkova, Irina; Raschel, Kilian. Random walks in $(\mathbb {Z}_{+})^{2}$ with non-zero drift absorbed at the axes. Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 3, pp. 341-387. doi : 10.24033/bsmf.2611. http://archive.numdam.org/articles/10.24033/bsmf.2611/

[1] P. Biane - « Quantum random walk on the dual of SU (n) », Probab. Theory Related Fields 89 (1991), p. 117-129. | MR | Zbl

[2] -, « Frontière de Martin du dual de SU (2) », in Séminaire de Probabilités, XXVI, Lecture Notes in Math., vol. 1526, Springer, 1992, p. 225-233. | Numdam | Zbl

[3] -, « Minuscule weights and random walks on lattices », in Quantum probability & related topics, QP-PQ, VII, World Sci. Publ., River Edge, NJ, 1992, p. 51-65. | MR | Zbl

[4] M. Bousquet-Mélou & M. Mishna - « Walks with small steps in the quarter plane », in Algorithmic probability and combinatorics, Contemp. Math., vol. 520, Amer. Math. Soc., 2010, p. 1-39. | MR | Zbl

[5] M.-F. Bru - « Wishart processes », J. Theoret. Probab. 4 (1991), p. 725-751. | MR | Zbl

[6] B. Chabat - Introduction à l'analyse complexe. Tome 1, Traduit du Russe: Mathématiques., “Mir”, 1990. | Zbl

[7] F. J. Dyson - « A Brownian-motion model for the eigenvalues of a random matrix », J. Mathematical Phys. 3 (1962), p. 1191-1198. | MR | Zbl

[8] P. Eichelsbacher & W. König - « Ordered random walks », Electron. J. Probab. 13 (2008), p. no. 46, 1307-1336. | EuDML | MR | Zbl

[9] G. Fayolle, R. Iasnogorodski & V. A. Malyshev - Random walks in the quarter-plane, Applications of Mathematics (New York), vol. 40, Springer, 1999. | MR | Zbl

[10] M. V. Fedoryuk - « Asymptotic methods in analysis », in Current problems of mathematics. Fundamental directions, Vol. 13 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1986, p. 93-210. | MR | Zbl

[11] D. J. Grabiner - « Brownian motion in a Weyl chamber, non-colliding particles, and random matrices », Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), p. 177-204. | EuDML | Numdam | MR | Zbl

[12] P.-L. Hennequin - « Processus de Markoff en cascade », Ann. Inst. H. Poincaré 18 (1963), p. 109-195. | EuDML | Numdam | MR | Zbl

[13] D. G. Hobson & W. Werner - « Non-colliding Brownian motions on the circle », Bull. London Math. Soc. 28 (1996), p. 643-650. | MR | Zbl

[14] I. Ignatiouk-Robert - « Martin boundary of a killed random walk on a half-space », J. Theoret. Probab. 21 (2008), p. 35-68. | MR | Zbl

[15] -, « Martin boundary of a reflected random walk on a half-space », Probab. Theory Related Fields 148 (2010), p. 197-245. | MR | Zbl

[16] I. Ignatiouk-Robert & C. Loree - « Martin boundary of a killed random walk on a quadrant », Ann. Probab. 38 (2010), p. 1106-1142. | MR | Zbl

[17] K. Johansson - « Shape fluctuations and random matrices », Comm. Math. Phys. 209 (2000), p. 437-476. | MR | Zbl

[18] -, « Non-intersecting paths, random tilings and random matrices », Probab. Theory Related Fields 123 (2002), p. 225-280. | MR | Zbl

[19] M. Katori & H. Tanemura - « Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems », J. Math. Phys. 45 (2004), p. 3058-3085. | MR | Zbl

[20] W. König & N. O'Connell - « Eigenvalues of the Laguerre process as non-colliding squared Bessel processes », Electron. Comm. Probab. 6 (2001), p. 107-114. | EuDML | MR | Zbl

[21] W. König, N. O'Connell & S. Roch - « Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles », Electron. J. Probab. 7 (2002), p. no. 5, 24 pp. | EuDML | MR | Zbl

[22] I. A. Kurkova & V. A. Malyshev - « Martin boundary and elliptic curves », Markov Process. Related Fields 4 (1998), p. 203-272. | MR | Zbl

[23] J. K. Lu - Boundary value problems for analytic functions, Series in Pure Mathematics, vol. 16, World Scientific Publishing Co. Inc., 1993. | MR | Zbl

[24] V. A. Malyshev - « An analytical method in the theory of two-dimensional positive random walks », Sib. Math. J. 13 (1972), p. 917-929. | Zbl

[25] -, « Asymptotic behavior of the stationary probabilities for two-dimensional positive random walks », Sib. Math. J. 14 (1973), p. 109-118. | MR | Zbl

[26] N. O'Connell - « Conditioned random walks and the RSK correspondence », J. Phys. A 36 (2003), p. 3049-3066. | MR | Zbl

[27] -, « A path-transformation for random walks and the Robinson-Schensted correspondence », Trans. Amer. Math. Soc. 355 (2003), p. 3669-3697. | MR | Zbl

[28] -, « Random matrices, non-colliding processes and queues », in Séminaire de Probabilités, XXXVI, Lecture Notes in Math., vol. 1801, Springer, 2003, p. 165-182. | Numdam | MR | Zbl

[29] N. O'Connell & M. Yor - « A representation for non-colliding random walks », Electron. Comm. Probab. 7 (2002), p. 1-12. | EuDML | MR | Zbl

[30] K. Raschel - « Random walks in the quarter plane absorbed at the boundary : Exact and asymptotic », preprint arXiv:0902.2785.

[31] G. Sansone & J. Gerretsen - Lectures on the theory of functions of a complex variable. II: Geometric theory, Wolters-Noordhoff Publishing, Groningen, 1969. | MR | Zbl

Cited by Sources: